Pitch evoked by spectral regularity (inharmonic components)

[Chen-gia TSAI <gia@xxxxxxxx>]

Hello everyone,

My study of the timbre of the Chinese flute leads me to find a strange effect of pitch sensation. I wonder if this effect can be explained by any model of pitch extraction.
In my experiment, all complex tones contain five components with frequencies {e, e+d, e+2d, e+3d, e+4d}. Although these spectral components are generally inharmonic, some pitches can be heard. As expected, one pitch is the frequency difference between the adjacent components; d. It may be evoked by DPOAEs.

Strangely enough, there is another pitch f.

For instance, a complex tone consisting of frequencies {1344, 1344+500, 1344+1000, 1344+1500, 1344+2000} evokes a pitch f = 336 Hz. Similarly, I found

{1680, 1680+500, 1680+1000, 1680+1500, 1680+2000} , f = 420 Hz
{2016, 2016+500, 2016+1000, 2016+1500, 2016+2000} , f = 504 Hz
{2285, 2285+500, 2285+1000, 2285+1500, 2285+2000} , f = 571.2 Hz

These four tones can be heard in the file  http://server37.hypermart.net/berlin-music/ex/a_500.wav.
In this file I demonstrate the pitch sensation f by pitch matching.

This kind of pitch sensation appears to be a function of the frequency of the lowest components e.
I get an empirical equation f = e*0.25 for 300 Hz < d < 1000 Hz and certain range of e.
I also synthesized files for d = 700 Hz and d = 300 Hz. The empirical equation seems to fail for d = 300 Hz (the last two tones).
http://server37.hypermart.net/berlin-music/ex/a_700.wav
http://server37.hypermart.net/berlin-music/ex/a_300.wav

I cannot explain this unexpected pitch sensation and the empirical equation. There are some papers dealing with the effect of grouping by spectral regularity, but I have not found studies on the pitch evoked by spectral regularity. It would be wonderful to have some literature to refer to this strange effect of pitch sensation, or some other ideas. Many thanks for any help!

All best,

Chen-Gia Tsai
gia@snafu.de
PhD Student (Musicology)